Using the fast Fourier transform to determine the period of a physical oscillator with precision

Measuring the period of torsion pendulums with precision has long been a formidable challenge in gravitation experiments, particularly those measuring the Newtonian gravitational constant G. An alternative method to fitting the position signal of the pendulum to a sine wave is the use of the power spectrum generated by the fast Fourier transform (FFT) as the source of information from which the period of oscillation can be determined. There are, however, known limitations to the use of a FFT to measure the period of a physical oscillator with precision. These limitations include two effects due to the finiteness of the duration of the sinusoidal data record and one effect due to the uncertainty of the starting phase of the oscillator relative to the window imposed by this duration. We have done a phenomenological study of the FFT using a desktop computer to imitate a precision oscillator having the physical characteristics of a finite damping constant and drift in the zero potential-energy position. Also, we have taken extensive data with a torsion pendulum, and analyzed them in this way. These studies show that for a real oscillator, such as the classical torsion pendulum, the FFT is a useful tool for determining the period of oscillation with the precision usually associated with larger, more complex, fitting algorithms. With good signal-to-noise ratio and under conditions appropriate to a torsion pendulum, the FFT method can measure the frequency or period to five parts in 106 or better.